Path integral of the hydrogen atom, the Jacobi's principle of least action and one-dimensional quantum gravity

A path integral evaluation of the Green's function for the hydrogen atom initiated by Duru and Kleinert is studied by recognizing it as a special case of the general treatment of the separable Hamiltonian of Liouville-type. The basic dynamical principle involved is identified as the Jacobi's principle of least action for given energy which is reparametrization invariant, and thus the appearance of a gauge freedom is naturally understood. The separation of variables in operator formalism corresponds to a choice of gauge in path integral, and the Green's function is shown to be gauge independent if the operator ordering is properly taken into account. Unlike the conventional Feynman path integral,which deals with a space-time picture of particle motion, the path integral on the basis of the Jacobi's principle sums over orbits in space. We illustrate these properties by evaluating an exact path integral of the Green's function for the hydrogen atom in parabolic coordinates, and thus avoiding the use of the Kustaanheimo-Stiefel transformation. In the present formulation , the Hamiltonian for Stark effect is converted to the one for anharmonic oscillators with an unstable quartic coupling. We also study the hydrogen atom path integral from a view point of one-dimensional quantum gravity coupled to matter fields representing the electron coordinates. A simple BRST analysis of the problem with an evaluation of Weyl anomaly is presented .